The following example uses the multiplication rule in combination with Equation 2-4 to answer a more diffi cult, but common, question. In random experiments in which items are selected from a batch, an item may or may not be replaced before the next one is selected. This is referred to as sampling with or without replacement, respectively.
The four white spaces occur between the fi ve black bars. In the fi rst step, focus on the bars. The number of permutations of fi ve black bars when two are B and three are b is
5 2 3 ! 10
! ! =
In the second step, consider the white spaces. A code has three narrow spaces w and one wide space W so there are four possible locations for the wide space. Therefore, the number of possible codes is 10 × 4 = 40. If one code is held back as a start/stop delimiter, then 39 other characters can be coded by this system (and the name comes from this result).
Sampling without Replacement A bin of 50 manufactured parts contains 3 defective parts and 47 nondefective parts. A sample of 6 parts is selected from the 50 parts without replacement. That is, each part can be selected only once, and the sample is a subset of the 50 parts. How many different samples are there of size 6 that contain exactly 2 defective parts?
A subset containing exactly 2 defective parts can be formed by fi rst choosing the 2 defective parts from the three defective parts. Using Equation 2-4, this step can be completed in
3 2
3
2 1 3
⎛
⎝⎜
⎞
⎠⎟ = ! =
! ! different ways
Example 2-14
FOR SECTION 2-1
Exercises
Provide a reasonable description of the sample space for each of the random experiments in Exercises 2-1 to 2-17. There can be more than one acceptable inter pretation of each experiment.
Describe any assumptions you make.
2-1. Each of three machined parts is classifi ed as either above or below the target specifi cation for the part.
2-2. Each of four transmitted bits is classifi ed as either in error or not in error.
2-3. In the fi nal inspection of electronic power supplies, either units pass, or three types of nonconformities might occur: func- tional, minor, or cosmetic. Three units are inspected.
2-4. The number of hits (views) is recorded at a high-volume Web site in a day.
2-5. Each of 24 Web sites is classifi ed as containing or not containing banner ads.
2-6. An ammeter that displays three digits is used to measure current in milliamperes.
2-7. A scale that displays two decimal places is used to meas- ure material feeds in a chemical plant in tons.
2-8. The following two questions appear on an employee survey questionnaire. Each answer is chosen from the fi ve- point scale 1 (never), 2, 3, 4, 5 (always).
Is the corporation willing to listen to and fairly evaluate new ideas?
How often are my coworkers important in my overall job performance?
2-9. The concentration of ozone to the nearest part per billion.
2-10. The time until a service transaction is requested of a computer to the nearest millisecond.
2-11. The pH reading of a water sample to the nearest tenth of a unit.
2-12. The voids in a ferrite slab are classifi ed as small, medium, or large. The number of voids in each category is measured by an optical inspection of a sample.
2-13. The time of a chemical reaction is recorded to the near- est millisecond.
2-14. An order for an automobile can specify either an automatic or a standard transmission, either with or without air condition- ing, and with any one of the four colors red, blue, black, or white.
Describe the set of possible orders for this experiment.
2-15. A sampled injection-molded part could have been pro- duced in either one of two presses and in any one of the eight cavities in each press.
2-16. An order for a computer system can specify memory of 4, 8, or 12 gigabytes and disk storage of 200, 300, or 400 gigabytes.
Describe the set of possible orders.
2-17. Calls are repeatedly placed to a busy phone line until a connection is achieved.
2-18. Three attempts are made to read data in a magnetic storage device before an error recovery procedure that reposi- tions the magnetic head is used. The error recovery procedure attempts three repositionings before an “abort’’ message is sent to the operator. Let
s denote the success of a read operation f denote the failure of a read operation
S denote the success of an error recovery procedure F denote the failure of an error recovery procedure A denote an abort message sent to the operator
Describe the sample space of this experiment with a tree diagram.
2-19. Three events are shown on the Venn diagram in the fol- lowing fi gure:
A B
C
Reproduce the fi gure and shade the region that corresponds to each of the following events.
(a) A′ (b) A∩B (c)
(
A∩B)
∪C (d)(
B∪C)
′ (e)(
A∩B)
′∪CProblem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
Then, the second step is to select the remaining 4 parts from the 47 acceptable parts in the bin. The second step can be completed in
47 4
47
4 43 178 365
⎛
⎝⎜
⎞
⎠⎟= ! =
! ! , different ways
Therefore, from the multiplication rule, the number of subsets of size 6 that contain exactly 2 defective parts is
3 178 365× , =535 095,As an additional computation, the total number of different subsets of size 6 is found to be
506 50
6 44 15 890 700
⎛
⎝⎜
⎞
⎠⎟= ! =
! ! , ,
Section 2-1/Sample Spaces and Events 27 2-20. Three events are shown on the Venn diagram in the fol-
lowing figure:
A B
C
Reproduce the figure and shade the region that corresponds to each of the following events.
(a) A′ (b)
(
A∩B)
∪(
A∩B′)
(c)
(
A∩B)
∪C (d)(
B∪C)
′(e)
(
A∩B)
′∪C2-21. A digital scale that provides weights to the nearest gram is used.
(a) What is the sample space for this experiment?
Let A denote the event that a weight exceeds 11 grams, let B denote the event that a weight is less than or equal to 15 grams, and let C denote the event that a weight is greater than or equal to 8 grams and less than 12 grams.
Describe the following events.
(b) A∪B (c) A∩B
(d) A′ (e) A∪ ∪B C
(f)
(
A∪C)
′ (g) A∩ ∩B C(h) B′ ∩C (i) A∪
(
B∩C)
2-22. In an injection-molding operation, the length and width, denoted as X and Y, respectively, of each molded part are eval- uated. Let
A denote the event of 48 < X < 52 centimeters B denote the event of 9 < Y < 11 centimeters
Construct a Venn diagram that includes these events. Shade the areas that represent the following:
(a) A (b) A∩B
(c) A′ ∪B (d) A∩B
(e) If these events were mutually exclusive, how successful would this production operation be? Would the process pro- duce parts with X=50 centimeters and Y = 10 centimeters?
2-23. Four bits are transmitted over a digi- tal communications channel. Each bit is either distorted or received without distortion. Let Ai denote the event that the ith bit is distorted, i= …1, ,4.
(a) Describe the sample space for this experiment.
(b) Are the Ai’s mutually exclusive?
Describe the outcomes in each of the following events:
(c) A1 (d) A1′
(e) A1∩A2∩A3∩A4 (f)
(
A1∩A2)
∪(
A3∩A4)
2-24. In light-dependent photosynthesis, light quality refers to the wavelengths of light that are important. The wavelength of a sample of photosynthetically active radiations (PAR) is measured to the nearest nanometer. The red range is 675–700 nm and the blue range is 450–500 nm. Let A denote the event
that PAR occurs in the red range, and let B denote the event that PAR occurs in the blue range. Describe the sample space and indicate each of the following events:
(a) A (b) B (c) A∩B (d) A∪B
2-25. In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified—one positive and one negative. Suppose that a replication is observed in three cells. Let A denote the event that all cells are identified as positive, and let B denote the event that all cells are nega- tive. Describe the sample space graphically and display each of the following events:
(a) A (b) B
(c) A∩B (d) A∪B
2-26. Disks of polycarbonate plastic from a supplier are analyzed for scratch and shock resistance. The results from 100 disks are summarized here:
Shock Resistance
High Low
Scratch High 70 9
Resistance Low 16 5
Let A denote the event that a disk has high shock resistance, and let B denote the event that a disk has high scratch resistance.
Determine the number of disks in A∩ ′B, A , and A∪B. 2-27. Samples of a cast aluminum part are classified on the basis of surface finish (in microinches) and edge finish. The results of 100 parts are summarized as follows:
Edge Finish
Excellent Good
Surface Excellent 80 2
Finish Good 10 8
(a) Let A denote the event that a sample has excellent sur- face finish, and let B denote the event that a sample has excellent edge finish. Determine the number of samples in A′∩ B, B′ and in A∪B.
(b) Assume that each of two samples is to be classified on the basis of surface finish, either excellent or good, and on the basis of edge finish, either excellent or good. Use a tree dia- gram to represent the possible outcomes of this experiment.
2-28. Samples of emissions from three suppliers are clas- sified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:
Conforms
Yes No
1 22 8
Supplier 2 25 5
3 30 10
Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to specifications.
Determine the number of samples in A′∩ B, B ,′ and A∪B.
2-29. The rise time of a reactor is measured in minutes (and fractions of minutes). Let the sample space be posi- tive, real numbers. Define the events A and B as follows:
A=
{
x x| < .72 5 and B}
={
x x| > .52 5 .}
Describe each of the following events:
(a) A′ (b) B′
(c) A∩B (d) A∪B
2-30. A sample of two items is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches:
(a) The batch contains the items { , , , }a b c d. (b) The batch contains the items { , , , , , , }a b c d e f g. (c) The batch contains 4 defective items and 20 good items.
(d) The batch contains 1 defective item and 20 good items.
2-31. A sample of two printed circuit boards is selected without replacement from a batch. Describe the (ordered) sample space for each of the following batches:
(a) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 2 boards with major defects.
(b) The batch contains 90 boards that are not defective, 8 boards with minor defects, and 1 board with major defects.
2-32. Counts of the Web pages provided by each of two com- puter servers in a selected hour of the day are recorded. Let A denote the event that at least 10 pages are provided by server 1, and let B denote the event that at least 20 pages are provided by server 2. Describe the sample space for the numbers of pages for the two servers graphically in an x−y plot. Show each of the following events on the sample space graph:
(a) A (b) B
(c) A∩B (d) A∪B
2-33. A reactor’s rise time is measured in minutes (and frac- tions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches. Let A denote the event that the rise time of batch 1 is less than 72.5 minutes, and let B denote the event that the rise time of batch 2 is greater than 52.5 minutes.
Describe the sample space for the rise time of two batches graphically and show each of the following events on a two- dimensional plot:
(a) A (b) B′
(c) A∩B (d) A∪B
2-34. A wireless garage door opener has a code deter- mined by the up or down setting of 12 switches. How many outcomes are in the sample space of possible codes?
2-35. An order for a computer can specify any one of five memory sizes, any one of three types of displays, and any one of four sizes of a hard disk, and can either include or not include a pen tablet. How many different systems can be ordered?
2-36. In a manufacturing operation, a part is produced by machining, polishing, and painting. If there are three machine tools, four polishing tools, and three painting tools, how many different routings (consisting of machining, followed by pol- ishing, and followed by painting) for a part are possible?
2-37. New designs for a wastewater treatment tank have proposed three possible shapes, four possible sizes, three loca- tions for input valves, and four locations for output valves. How many different product designs are possible?
2-38. A manufacturing process consists of 10 operations that can be completed in any order. How many different pro- duction sequences are possible?
2-39. A manufacturing operation consists of 10 opera- tions. However, five machining operations must be completed before any of the remaining five assembly operations can begin.
Within each set of five, operations can be completed in any order. How many different production sequences are possible?
2-40. In a sheet metal operation, three notches and four bends are required. If the operations can be done in any order, how many different ways of completing the manufacturing are possible?
2-41. A batch of 140 semiconductor chips is inspected by choosing a sample of 5 chips. Assume 10 of the chips do not conform to customer requirements.
(a) How many different samples are possible?
(b) How many samples of five contain exactly one noncon- forming chip?
(c) How many samples of five contain at least one noncon- forming chip?
2-42. In the layout of a printed circuit board for an electronic product, 12 different locations can accommodate chips.
(a) If five different types of chips are to be placed on the board, how many different layouts are possible?
(b) If the five chips that are placed on the board are of the same type, how many different layouts are possible?
2-43. In the laboratory analysis of samples from a chemical process, five samples from the process are analyzed daily. In addition, a control sample is analyzed twice each day to check the calibration of the laboratory instruments.
(a) How many different sequences of process and control sam- ples are possible each day? Assume that the five process samples are considered identical and that the two control samples are considered identical.
(b) How many different sequences of process and control sam- ples are possible if we consider the five process samples to be different and the two control samples to be identical?
(c) For the same situation as part (b), how many sequences are possible if the first test of each day must be a control sample?
2-44. In the design of an electromechanical product, 12 com- ponents are to be stacked into a cylindrical casing in a manner that minimizes the impact of shocks. One end of the casing is designated as the bottom and the other end is the top.
(a) If all components are different, how many different designs are possible?
(b) If seven components are identical to one another, but the others are different, how many different designs are possible?
(c) If three components are of one type and identical to one another, and four components are of another type and identical to one another, but the others are different, how many different designs are possible?
2-45. Consider the design of a communication system.
(a) How many three-digit phone prefixes that are used to repre- sent a particular geographic area (such as an area code) can be created from the digits 0 through 9?
Section 2-1/Sample Spaces and Events 29 (b) As in part (a), how many three-digit phone prefixes are possible
that do not start with 0 or 1, but contain 0 or 1 as the middle digit?
(c) How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?
2-46. A byte is a sequence of eight bits and each bit is either 0 or 1.
(a) How many different bytes are possible?
(b) If the first bit of a byte is a parity check, that is, the first byte is determined from the other seven bits, how many different bytes are possible?
2-47. In a chemical plant, 24 holding tanks are used for final product storage. Four tanks are selected at random and without replacement. Suppose that six of the tanks contain material in which the viscosity exceeds the customer requirements.
(a) What is the probability that exactly one tank in the sample contains high-viscosity material?
(b) What is the probability that at least one tank in the sample contains high-viscosity material?
(c) In addition to the six tanks with high viscosity levels, four different tanks contain material with high impurities. What is the probability that exactly one tank in the sample con- tains high-viscosity material and exactly one tank in the sample contains material with high impurities?
2-48. Plastic parts produced by an injection-molding operation are checked for conformance to specifications. Each tool contains 12 cavities in which parts are produced, and these parts fall into a conveyor when the press opens. An inspector chooses 3 parts from among the 12 at random. Two cavities are affected by a temperature malfunction that results in parts that do not conform to specifications.
(a) How many samples contain exactly 1 nonconforming part?
(b) How many samples contain at least 1 nonconforming part?
2-49. A bin of 50 parts contains 5 that are defective. A sample of 10 parts is selected at random, without replacement. How many samples contain at least four defective parts?
2-50. The following table summarizes 204 endothermic reac- tions involving sodium bicarbonate.
Final Temperature
Conditions Heat Absorbed (cal) Below Target Above Target
266 K 12 40
271 K 44 16
274 K 56 36
Let A denote the event that a reaction’s final temperature is 271 K or less. Let B denote the event that the heat absorbed is below target. Determine the number of reactions in each of the following events.
(a) A∩B (b) A′ (c) A∪B (d) A∪ ′B (e) A′∩B′
2-51. A Web ad can be designed from four different colors, three font types, five font sizes, three images, and five text phrases. How many different designs are possible?
2-52. Consider the hospital emergency department data in Exam- ple 2-8. Let A denote the event that a visit is to hospital 1, and let B denote the event that a visit results in admittance to any hospital.
Determine the number of persons in each of the following events.
(a) A∩B (b) A′ (c) A∪B (d) A∪ ′B (e) A′∩B′
2-53. An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7, pp. 111–127)] provided the following table of well failures for different geological formation groups in Baltimore County.
Wells Geological Formation Group Failed Total
Gneiss 170 1685
Granite 2 28
Loch raven schist 443 3733
Mafic 14 363
Marble 29 309
Prettyboy schist 60 1403
Other schists 46 933
Serpentine 3 39
Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed.
Determine the number of wells in each of the following events.
(a) A∩B (b) A′ (c) A∪B (d) A∩ ′B (e) A′∩B′ 2-54. Similar to the hospital schedule in Example 2-11, sup- pose that an operat ing room needs to handle three knee, four hip, and five shoulder surgeries.
(a) How many different sequences are possible?
(b) How many different sequences have all hip, knee, and shoulder surgeries scheduled consecutively?
(c) How many different schedules begin and end with a knee surgery?
2-55. Consider the bar code in Example 2-12. One code is still held back as a delimiter. For each of the following cases, how many characters can be encoded?
(a) The constraint of exactly two wide bars is replaced with one that requires exactly one wide bar.
(b) The constraint of exactly two wide bars is replaced with one that allows either one or two wide bars.
(c) The constraint of exactly two wide bars is dropped.
(d) The constraints of exactly two wide bars and one wide space are dropped.
2-56. A computer system uses passwords that contain exactly eight characters, and each character is 1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9).
Let Ω denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Determine the number of pass- words in each of the following events.
(a) Ω (b) A (c) A′∩B′ (d) Passwords that contain at least 1 integer (e) Passwords that contain exactly 1 integer
2-57. The article “Term Efficacy of Ribavirin Plus Interferon Alfa in the Treat ment of Chronic Hepatitis C,” [Gastroenterol- ogy (1996, Vol. 111, no. 5, pp. 1307–1312)], considered the effect of two treatments and a control for treatment of hepatitis C. The following table provides the total patients in each group and the number that showed a complete (positive) response after 24 weeks of treatment.